Approach to Equi/i6rium1 JOEL H. HILDEBRAND University of California Berkeley, California
T
HERE seems to be no topic in freshman chemistry that presents more difficulties to students than chemical equilibrium. After trying for over 30 years to give clear answers to their questions, I have come to have a great deal of sympathy with them, realizing that the subject is inherently a difficult one. There is evidence that i t sometimes bothers instructors and even writers of textbooks. Professional courtesy prevents giving examples. One may, of course, announce rules and formulas and drill students in applying them, but the teacher who does that is evading his obligation to assist his students to gain true understanding. In an effort to treat some of the questions of practical pedagogy involved, let'us discuss several examples of typical student difficulties. We inform the student that the proper way to write the solubility product for Ca(OHj2is K3 = (Ca++)(OH-j2,but why, he asks, is it (OH-j2? The teacher may reply that since the equation For solution is Ca(OH)2= Ca++ 2OH-, tlie product of the three ions-ou the right must be used, (Ca++) (OH-)(OH-). If the student accepts this explanation, he may then ask the reasonable question, as many of mine have, whether in a saturated solution of Ca(OH)2 in pure water, where (Ca++) = 0.02, and (OH-) = 0.04, one should write accordingly & = 0.02 X 0.02 X 0.02? Any instructor can answer dogmatically, "No, But can he make the reason one writes 0.02 X for the latter choice clear? Or, again, if the student should ask, as he may, "Why is the equilibrium pressure of COz over a mixture of CaC03 and CaO independent of the rglative amounts of the two solid phases?" what is the instructor to answer? Or, to consider a third case, how many instructors can themselves give a satisfactory explanation of the fact that although carbonic acid is weaker than acetic, a molar solution of sodium bicarbonate is less alkaline than molar sodium acetate? The basis of such difficulties lies, I believe, in the statistical basis of equilibrium; the student has to cultivate an imagination able to visualize t$e simultaneous, different behaviors of molecules of several diierent species. This is much harder, both to see and to illustrate, than the reactions of individual molecules. It would be easier it simple probability theory had a larger place in elementary mathematics. This could lend interest to that subject since so many human problems involve degrees of probability rather than true or false logical propositions.
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m2."
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1 Presented before the Division of Chemical Education at the 110th meeting of the American Chemical Society in Chicago, September 9-13. 1946.
The instructor should, of course, be able to state clearly the actual processes going on in a dynamic equilibrium, such as that between acetic acid and its ions in aqueous solution, but teachers should never forget that everybody finds it difficult to solve problems in unfamiliar terms; i t is in practice easier to compute the price of five oranges a t 36 cents per dozen than the price of five squirnks if 12 of them cost 36 shekels. Few of us would be quite so helpless as the poor woman on the radio quiz program who excused herself from naming a "city in Europe directly south of the North Pole" on the grounds that she had "never been to Europe," but we are all a little like that; most chemists would tackle a problem in which t stands for temperature with far more confidence than if it stands for time, so let us help out our struggling freshmen wherever possible, even though our best illustrations must admittedly be imperfect. A dance offers a partial analogy with a chemical equilibrium, such as the ionization of a weak acid. If HC2H302 H + C2H3Os: is-a little mysterious2 and unreal durinethe first term of the freshman year, gent. lady gent.-lady (dancing couple) is familiar enough-to most. The reaction takes place in "both directions" more or less simultaneously. There is "disordered motion," both linear apd rotational, and energy may even be expended in vibration. There are, a t any instant, a certain number of "wall flowers," or lone "ions," but not always the same individuals, and their number will vary with the number of opposite partners present; if more "gents" arrive, the number of unpaired ladies diminishes, but the type of the reaction is not altered, only the number of couples. One never lady $ (gent.)%-lady,no sees the reaction: 2 gents. matter how many of the former are present. This is worth carefully noting since students, if asked to represent what happens when sodium acetate is added to the acetic acid equilibrium, often add C2H302-not t o the solution but to one side of the equation, thereby spoiling it as an equation. The effectof diluting acetic acid can be illustrated by throwing open an adjacent ballroom, permitting a crowded dance to spread over more space. I n the interchange between dances partners will then be harder to locate, and the percentage of "ions" will be, temporarily, Ingenious instructors can think of .. neater. -
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+
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An instructor who insists that hydrogen ion is definitely H?O+ a
must use a different and a simpler process to fit this illustratmn.
other ways of displacing the equilibrium, e. g., by the escape of a number of individuals of one of the species to consult the mirror in the dressing room. The illustration can be only qualitative, but it is very important, in my opinion, for the student to get a clear qualitative concept of equilibrium before resorting to quantitative formulations. Not all will agree with this; orthodox instruction in physics nearly always consists in deriving formulas and turning them over to the student to apply, often in a rather mechanical way. I regard this as, first of all, poor pedagogy, for our object should not be limited to getting the right answers; second, impractical, since we want qualitative answers far more often than quantitative; and, third, misleading, sidce the theory is often qualitatively correct but quantitatively very inaccurate, as always where concentrations of ions are used in place of their more accurate activities. But to return to the solubility product of Ca(OH)Z. The solid in equilibrium with its saturated solution is always dissolving a t one point and being reformed a t another, as shown by the growth of fine crystals. The redeposition involves the union of one Ca++ and two OH-. If the concentration'of OH- is doubled, the rate of deposition is quadrupled until the concentration has become one-fourth as large, thus restoring the rate to equal again the rate of solution. But ions are difficult to imagine, so let us consider hunters out to shoot deer in a large area of some National Forest. Each hunter is allowed to shoot two deer. How will the rate a t which hunters get their "limits" depend on the "concentration" of the deer in the region? It is obviously proportional to the "concentration" of hunters. If n is the average number of deer per square mile, the speed with which a hunter locates and shoots his first deer is proportional to n. There are then n - 1 deer left, so that the s ~ e e dwith which he gets his second deer is urouor. . tional to n - 1. He gets his pair at a rate proportional to n(n - 1). If n is large, this is practically n2. It is not (n/2) (n/2); the chance of gettingmne deer is dependent on the total number of deer still alive, since any one of them will serve the purpose, but shooting one does little to decrease the chance of getting within range of the second. It is not necessary, we see, to have
a "triple collision," which is very improbable, in order for the law to hold, for A B -+AB followed by AB B -+ ABa gives also the statistical probability corresponding to ndnB2. Let us consider, finally, the problem of the alkalinity of a solution of sodium bicarbonate; why is it less than one of sodium acetate? There are two ways of representing the true situation. If HC03- behaved only in its one role as the ion of a weak acid, uniting with some of the H + of water, thus,
+
HCOs-
+
+ HzO = HICOa + OH-
the OH- would be liberated to a higher concentration than in the case of acetate ion, as we can infer from the fact that plain soda water is far less sour than vinegar; but HC03- has a second role. It is not only the ion of the weak acid, H2C03,but it is itself an acid, dissociating thus: HCOI- = COa--
+
+H
+
But since H+ OH- = HzO, these two processes assist each other, giving as a net result 2HC03- = HKOa
+ C0,--
with HzCOa and C03-- nearly, but not quite, equal, since an indicator test shows the concentration of OHin 1 M NaHC03 to be 2 X IO-= instead of IO-'. Now let us put the case a little differently. Among a lot of bicarbonate ions, as here represented, one may lose its H C which may later attach itself to one of the HCOa-, since there are so many of them, rather than rejoin the C03--, in spite of the greater attraction of the latter when on equal terms.
,..
H'eoa-
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It is, of course, desirable to havk.an idea of the concentrations present in such a solution as well as in solutions of COz and Na2COs. The following summary is based upon experimental data for 25' readily verifiable. Solution Con at 1 atm.
+ CO$ = HzCOs = H + + HCOa0.034 M
1 M NaHCOa 0 5 M NasCOa
1. CHANGE IN PH ON ADDINGTO 1 LITEROF 1 M FIGURE NAHCOI, FRACTIONS OF A LITER 01. (a) 1 M NAOH (TO THE LEFTOF 0). AND (b) 1 M HCL (TO THE RIGHT)
Present at equilibria
H.0
0
lo-'
2HCOa- = HKOa 0.98 M lo-' C08-H 2 0 = HCOs0.49 M
+
(since
+ COa-. +IO-'OH-
Nature has been considerate in making four of these concentrations all equal to 10W2. We can now ask a number of interesting questions involving shifts in these equilibria, such as the effectof adding Ca++, boiling, diluting, etc. After, but not before, developing skill in handling and understanding such qualitative questions, it is time for students who are not appalled by a little algebra to learn something about quantitative treatment. The first ionization constant of carbonic add can be calcu-
ated from the first set of experimental figures above.
More accurate data raise this to 4.3 X lo-', but the difference has little practical significance when temperature is not narrowly controlled. The ionization constant for HCOa-, K2,can be calculated from the above figures for 0.5 M Na2C03. Its value is to be found by inserting the proper figures for the concentrations contained in the expression:
These values are as follows: (H+) = 10-"/(OH-)
=
(C03--) = 0.5, since only one per cent of it is changed to HCOI-. (HCOa-) = (OH-) = lo-=, since the hydrolysis equation shows that one OH- IS liberated for every HC03- formed. Therefore,
Note that i t is not necesssary to introduce the constant for the hydrolysis equation as some teachers feel they must do. Having K1 and Kz, i t is now easy to show that they yield the figures given for the equilibrium 2HCO2- = C W HxCOZ
+
for
Since (HaC08)and (COa--) are nearly equal, each is 0.8 X 10P2when (HCOJ-) = 1: There are always a few intelligent students in every class whose appetite for chemistry is almost unlimited, and it is important to offer them plenty of solid food. Here are a few suggestions. 1. Reproduce the curve shown in Figure 1 by plotting a series of values of pH calculated by the aid of K1 and K2 for various ratios of (HzCO3)to (HC03-) and (HCO1-) to (COa--) for the respective branches. To get the value for pure 1 M HC05- shown in the curve, it is necessary to note that (COa--) is not exactly equal to (H,CO,), as assumed above, but rather that (H+)
+ (COz--)
=
(OH-)
+ (H*COd
2. Calculate the ratio (HaCOa)/(HCO3-) in blood from its pH = 7.4. 3. The solubility product of CaCOa is 5X10-'. The partial pressure of COz in air is 3 X lo-' atmospheres. From these figures and data selected from the above, calculate (a) the number of grams of CaC08 that could be dissolved by one cubic meter of natural surface water flosing over limestone rock, and (b) the
number of grams of soap, considered as ClsH3tO?Na, that this hard water could destroy. 4. The value of (Ca++)in seawater is approximately 10F2M. What pH must an oyster be able to establish in order to makes its shell from this Ca++ and the COa dissolved in sea water? (Cf. Problem 3.) This is a hard one, but a student of chemistry should try to know as much as an oyster about pH. 5. Clinch the whole drill by applying it to Has, with Kl = lo-', Ka = 10-15, and (H2S) = 0.1 a t 1 atmosphere and room temperature. Calculate (S--) for H 9 in the presence of 0.1 M HCl, consult solubility products for FeS and CuS, and see why the latter is precipitated but not the former. ina all;, explain why HgS dissolves to form HgSa-- in a solution of sodiun~ sulfidebut not in one of "ammonium sulfide." By this time, the student will know a lot about chemical equilibrium. Now, finally, I should like to inject something controversial. Some may object to my presentation in that I have not used what many regard as the "modern" and only correct svstem of acids and bases. I should bave written not H+, an abbreviation for H + (aq.), but H30+, and I should not havementioned the taboo word, "hydrolysis." My defense is that any system of acids and bases represents not truth but convenience, appropriateness to the particular phenomena under consideration. These may involve solutions in water, in liquid ammonia, or in the favorite solvents of the organic chemist, or they may involve molten lavas. One should choose the system most appropriate to the case, just as the musician choosesthe clef corresponding to the range ofthe particular .instrument. My late distinguished colleague, Gilbert Lewis, stated clearly, as early as 1923, the different possible acid-base systems, including the one now usually called the Bransted system. The advantages of this system for the ordinary purposes of the organic chemist are so great as to bave caused a wave'of enthusiasm and to have attracted adherents who insist that i t is the one modern, correct system, and must be taught as such even to freshmen. Evidently not all have appreciated the experiments of Lewis in which typical acid-base reactions and indicator changes take place with acids and solvents which can donate no protons to bases. Surely all should conclude from this that the Bronsted is not the one and only admissible system. I, myself, fully appreciate, I trust, its usefulness for the purposes for which it is appropriate, but I do not use i t in teachina freshmen for two reasons: first, I reaard i t as better pedagogy to begin with the ordinary-water system, expanding to the proton-donor-acceptor system later for the special purposes of organic chemistry; and second, because the other systems are equally "right" and so much more useful for certain purposes tbat the Bronsted viewpoint should not be allowed to usurp the entire field. A teacher in another institution has stated his experience regarding the pedagogic difficulties of this approach as follows: '
At our institution I have been required t o present the Briinsted definitions as the definitions of acids and bases. T o avoid confusion and doubts, the students are not supposed t o be made aware of the existence of other points of view. The students have great difficulty with the Brdnsted concept. We make the presentation as simde as possible, then conclude that the students haven't grasped it, sowe go over it again and again. We spend an inordinate amount of time on i t a n d then recognize that the students have digested very little of it, so we ask practically nothing about it in the examinations. Far the sake of simplicity we practically ignore the Briinsted concepts in the remainder of the course. All in all, the result is a mess. I believe the majority of our freshman stall agrees with me in this conclusion, and I believe the Briinsted theory is ignored completely in the analytical courses that follow. I am decidedly not in favor of aoina . .Br6nsted in the freshman courses.
The "confirmed Bronstedians," who, would criticize me for writing H + (aq.), or H + for short, are not, I find, always consistent with respect to other ions. For example, Sn++++in aqueous solution is as assuredly an acid as is "H30+"; why should the water not be in.dicated? The reason is, I suspect, that they do not :know exactly how much water is attached to stannic :ion, and they know that they do not know. They should realize that they are equally ignorant regarding
the hydration of hydrogen ion; with hydrogen bonds being formed and broken continually in water, how can anyone be sure that a proton isolates and attaches itself to only one water molecule? I note a further illusion-namelv.,, that the writinc e of an equilibrium in the form, acid = proton conjugate base, allows us to express the true strengths of acids, apart from the solvents used. This is only roughly true, because the particular solvent used affects not only the proton but the activities of the acid and the base as well. There is no such thing as a true constant for such an equilibrium divorced from the solvent. I hope not to be misunderstood as objecting to the value and use of the Bronsted concepts, partic;larly in organic chemistry. What I criticize is only, first, the notion that they alone are right and, second, the poor pedagogy of beginning the teaching of acid-base equilibria in these terms. In the forthcoming, B t h edition of my "Principles of Chemistry," I have endeavored to give, in a whole chapter devoted to "Systems of Acids and Bases," as full and sympathetic account of the Bronsted system as possible, and I should regret being considered as an out-and-out "anti-Bronstedian."
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